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A115029 Number of partitions of n such that all parts, with the possible exception of the smallest, appear only once.

%I A115029 #29 Jun 15 2025 03:06:20
%S A115029 1,1,2,3,5,6,10,12,17,22,29,36,48,59,73,93,114,139,171,207,250,304,
%T A115029 361,432,517,613,722,856,1005,1178,1382,1612,1875,2184,2528,2927,3386,
%U A115029 3900,4486,5159,5916,6772,7749,8843,10078,11482,13048,14811,16805,19026
%N A115029 Number of partitions of n such that all parts, with the possible exception of the smallest, appear only once.
%C A115029 Also number of partitions of n such that if k is the largest part, then k and all integers from 1 to some integer m, 0<=m<k, occur any number of times (if m = 0, then partition consists only of k's). Example: a(5)=6 because we have [5], [4,1], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1] ([3,2] does not qualify). - _Emeric Deutsch_, Apr 19 2006
%H A115029 Alois P. Heinz, <a href="/A115029/b115029.txt">Table of n, a(n) for n = 0..10000</a>
%F A115029 G.f.: 1+Sum_{k>=1} x^k/(1-x^k)*Product_{i>=k+1} (1+x^i).
%F A115029 G.f.: 1+Sum_{k>=1} (x^k/(1-x^k)) * Sum_{m=0..k-1} x^(m*(m+1)/2) / Product_{i=1..m} (1-x^i). - _Emeric Deutsch_, Apr 19 2006
%F A115029 a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - _Vaclav Kotesovec_, Jun 15 2025
%e A115029 a(5) = 6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1] ([2,2,1] does not qualify).
%p A115029 g:=1+sum(x^k/(1-x^k)*product(1+x^i,i=k+1..90),k=1..90): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..44); # _Emeric Deutsch_, Apr 19 2006
%p A115029 # second Maple program:
%p A115029 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, b(n, i-1)+
%p A115029       `if`(irem(n, i)=0, 1, 0)+`if`(n>i, b(n-i, i-1), 0))
%p A115029     end:
%p A115029 a:= n-> b(n$2):
%p A115029 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 03 2019
%t A115029 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[Mod[n, i] == 0, 1, 0] + If[n > i, b[n - i, i - 1], 0]];
%t A115029 a[n_] := b[n, n];
%t A115029 a /@ Range[0, 50] (* _Jean-François Alcover_, Nov 21 2020, after _Alois P. Heinz_ *)
%Y A115029 Cf. A034296.
%K A115029 easy,nonn
%O A115029 0,3
%A A115029 _Vladeta Jovovic_, Feb 25 2006; corrected Mar 05 2006
%E A115029 a(0)=1 prepended by _Alois P. Heinz_, Feb 03 2019